While the motives behind the incidents -- a knife attack on police and two car rampages onto passers-by -- remain unclear, the violence has jarred nerves after repeated jihadist calls for "lone wolf" action in France over its fight against Islamic extremism.

"Fear over Christmas" titled local daily Le Parisien, while Le Figaro newspaper wrote a front-page editorial headlined "enemies from within."

The violence began on Saturday when a man reportedly shouting "Allahu Akbar" was shot dead after walking into a police station in the central town of Joue-les-Tours and attacking three officers with a knife, two of whom were seriously injured.

Then on Sunday evening, a driver ploughed into pedestrians in Dijon in the east, injuring 13 people and also shouting the same Islamic phrase which means "God Is Greater" and has in the past been used by extremists when waging violent attacks.

And on Monday night, another man rammed into a bustling Christmas market with his car in the western city of Nantes, injuring 10 people -- one of whom is critical -- before stabbing himself repeatedly and being arrested.

- 'No link' between attacks -

The attacks both differ widely and present disturbing similarities, and while Prime Minister Manuel Valls stressed Tuesday there was "no link" between the three, the ever-present threat of attacks inspired by Islamic extremism has authorities concerned.

In September, the radical Islamic State group that controls swathes of Iraq and Syria urged Muslims around the world to kill "in any manner" those from countries involved in a coalition fighting its jihadists, singling out the French.

Among instructions detailing how to kill civilians or military personnel was to "run him over with your car."

But while the probe into Saturday's attack is veering towards extremism -- the Burundian convert to Islam who attacked police had posted an Islamic State flag on his Facebook page -- the last two car rampages appear to have been committed by "unbalanced" people.

Both prosecutors in charge of probing the driving incidents insisted they were not "terrorist acts".

The assailant in Dijon, for instance, had been to psychiatric hospital 157 times, local prosecutor Marie-Christine Tarrare told reporters.

She said he told police that he ploughed into people due to a sudden "outburst of empathy for the children of Chechnya" and had shouted "Allahu Akbar" to give him courage.

Interior Minister Bernard Cazeneuve meanwhile said the attacker in Nantes also appeared to be "unbalanced" and not motivated by politics or religion.

A source close to the investigation said that after slamming into shoppers, the driver stabbed himself in the chest "at least nine times", causing himself serious -- but not life-threatening -- injuries.

- Minimising risk? -

Nevertheless, the government faced criticism Tuesday that it was minimising the threat, at a time when more than 1,000 nationals are involved in one way or another with jihad in Syria or Iraq -- be it on home soil or in both countries.

Saturday's assailant Bertrand Nzohabonayo was not on a domestic intelligence watch-list but his brother Brice is well known for his radical views and was arrested in Burundi soon after the incident.

Nzohabonayo's mother had also told authorities that she was worried about Brice's radicalisation and "the influence he could have on his brother Bertrand," said Paris prosecutor Francois Molins, whose office is in charge of the probe.

The assailant in Dijon, meanwhile, had taken an interest in religion and started wearing a djellaba -- a long robe worn in Muslim countries -- just a week ago, according to his mother.

Valls said he was meeting with his ministers on Tuesday to discuss the situation.

"The best response is to continue to live peacefully with the necessary vigilance of course," he told French radio.